Advanced mathematics is largely about logical argument, as much as it is computation or calculation. Over time, as each generation extended their ideas into new realms, they looked at the logical arguments of their predecessors and found that there were gaps, elisions, things that were not fully understood. One could imagine that this process might continue forever, but it does not: in the early twentieth century, a number of mathematicians (most notably Hilbert) completed the project of “hitting bedrock”, finding a clear demarcation line between certainty and uncertainty. This required a clarification about what mathematics and reasoning are about (“formalism”). These profound ideas deserve to be better known outside of professional mathematics. (Later, some mathematicians (notably Gödel) used this framework to show the limitations of the framework, in a different and more precise sense than before.) In this class we will go through the development of these ideas. No mathematical knowledge will be assumed.

(May 8, 12, 15, 19, 22, 26)